Idea

A diffeological spaces is a type of generalized smooth space. As with the other variants, it subsumes the notion of smooth manifold but also naturally captures other spaces that one would like to think of as smooth spaces but aren’t manifolds; for example, the space of all smooth maps between two smooth manifolds can be made into a diffeological space. (These mapping spaces are rarely manifolds themselves, see manifolds of mapping spaces.)

In a little more detail, a diffeology, 𝒟\mathcal{D} on a set XX is a presheaf on the category of open subsets of Euclidean spaces with smooth maps as morphisms. To each open set U⊆ℝnU \subseteq \mathbb{R}^n, it assigns a subset of Set(U,X)\Set(U,X). The functions in Set(U,X)\Set(U,X) are to be regarded as the “smooth functions” from UU to XX. A diffeological space is then a set together with a diffeology on it.

A morphism of diffeological spaces is a morphism of the corresponding sheaves: we take DiffeologicalSp↪Sh(CartSp)DiffeologicalSp \hookrightarrow Sh(CartSp) to be the full subcategory on the diffeological spaces in the sheaf topos.

For (X,𝒟)(X,\mathcal{D}) a diffeological space, and for any U∈𝒪𝓅U \in \mathcal{Op}, the set 𝒟(U)\mathcal{D}(U) is also called the set of plots in XX on UU. This is to be thought of as the set of ways of mapping UU smoothly into the would-be space XX. This assignment defined what it means for a map U→XU \to X of sets to be smooth.

A sheaf on the site 𝒪𝓅\mathcal{Op} of open subsets of Euclidean spaces is completely specified by its restriction to CartSp, the full subcategory of Euclidean spaces. Therefore in the sequel we shall often restrict our attention to CartSp.

One may define a “very general smooth space” to be any sheaf of CartSp and identify the sheaf toposSh(CartSp)Sh(CartSp) as the category of very general smooth spaces.

The concreteness condition on the sheaf is a reiteration of the fact that a diffeological space is a subsheaf of the sheaf U↦X|U|U \mapsto X^{|U|}. In this way, one does not have to explicitly mention the underlying set XX as it is determined by the sheaf on the one-point open subset of ℝ0\mathbb{R}^0.

Examples

Every smooth manifoldXX, i.e. every object of Diff, becomes a diffeological space by defining the plots on U∈CartSpU \in CartSp to be the ordinary smooth functions from UU to XX, i.e. the morphisms in Diff:

X:U↦HomDiff(U,X).
X : U \mapsto Hom_{Diff}(U,X)
\,.

For XX and YY two diffeological spaces, their product as sets X×YX \times Y becomes a diffeological space whose plots are pairs consisting of a plot into XX and one into YY

Given any two diffeological spaces XX and YY, the set of morphisms HomDiffSp(X,Y)Hom_{DiffSp}(X,Y) becomes a smooth space by taking the plots on some UU to be the smooth morphisms X×U→YX \times U \to Y, i.e. the smooth UU-parameterized families of smooth maps from XX to YY:

Proof

To see that the functor is faithful, notice that if f,g:X→Yf,g : X \to Y are two smooth functions that differ at some point, then they must differ in some open neighbourhood of that point. This open ball is a plot, hence the corresponding diffeological spaces differ on that plot.

To see that the functor is full, we need to show that a map of sets f:X→Yf : X \to Y that sends plots to plots is necessarily a smooth function, hence that all its derivatives exist. This can be tested already on all smooth curves γ:(0,1)→X\gamma : (0,1) \to X in XX. By Boman's theorem, a function that takes all smooth curves to smooth curves is necessarily a smooth function. But curves are in particular plots, so a function that takes all plots of XX to plots of YY must be smooth.

Remark

The proof shows that we could restrict attention to the full sub-site CartSpdim≤1⊂CartSpCartSp_{dim \leq 1} \subset CartSp on the objects ℝ0\mathbb{R}^0 and ℝ1\mathbb{R}^1 and still have a full and faithful embedding

While the site CartSpdim≤1CartSp_{dim \leq 1} is more convenient for some purposes, it is not so useful for other purposes, mostly when diffeological spaces are regarded from the point of view of the full sheaf topos: the sheaf topos Sh(CartSpdim≤1)Sh(CartSp_{dim \leq 1}) lacks some non-concrete sheaves of interest, such as the sheaves of differential forms of degree ≥2\geq 2.

Embedding of diffeological spaces into the topos of smooth spaces

We discuss aspects of the full sheaf toposSh(CartSp)Sh(CartSp) on the siteCartSp – the topos of smooth spaces – and of how diffeological spaces are embedded into this. In summary, we have that Sh(CartSp)Sh(CartSp) is a cohesive topos and that DiffeologicalSpace↪Sh(CartSp)DiffeologicalSpace \hookrightarrow Sh(CartSp) is the canonical sub-quasitopos of concrete sheaves inside it.

Proposition

Proof

The following argument works for every siteCC which is such that constant presheaves on CC are already sheaves.

Notice that this is the case for C=CartSpC = CartSp because every Cartesian space is connected: for S∈SetS \in Set a compatible family of elements of ConstSConst S on a cover {Ui→ℝn}\{U_i \to \mathbb{R}^n\} of some ℝn\mathbb{R}^n is an element of SS on each patch, such that their restriction maps to intersections of patches coincide. But the restriction maps are all identities, so this says that all these elements coincide. Therefore the set of compatible families is just the set SS itself, hence the presheaf ConstSConst S is a sheaf.

So with L:PSh(C)→Sh(C)L : PSh(C) \to Sh(C) the sheafification functor we have that LConstS≃ConstSL Const S \simeq Const S.

Whenever this is the case the left adjoint to the constant presheaf functor, which always exists for presheaves and is given by the colimit functor, is also left adjoint on the level of sheaves, because for each X∈Sh(C)X \in Sh(C) and S∈SetS \in Set we have natural bijections

Proposition

Proof

Since CartSpCartSp is a connected category it is immediate that Const:Set→PSh(CartSp)Const \colon Set \to PSh(CartSp) is a full and faithful functor. By the above this equals LConstL Const, which is hence also full and faithful.

By the discussion at connected topos we could equivalently convince ourselves that Π0\Pi_0 preserves the terminal object. The terminal object of Sh(CartSp)Sh(CartSp) is y(ℝ0)y(\mathbb{R}^0), hence representable. By the above, Π0\Pi_0 sends all representable objects to the singleton set, which is the terminal object of SetSet.

that takes a Cartesian space UU to the set of functions from its underlying set of points to SS. This is clearly a sheaf (a function of sets from UU to SS is clearly fixed by all its restrictions to a collections of subsets of UU whose unition is UU.)